3.31.45 \(\int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [3045]

3.31.45.1 Optimal result

Integrand size = 62, antiderivative size = 455 \[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\left (32760 b^2 c^2-1024 b c^6+3465 a b^2 x-1920 a b c^4 x+114240 a^2 b c^2 x^2-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3+40320 a^4 c^2 x^4\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-2310 b^2 c+768 b c^5+38640 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (3465 b^2-640 b c^4+94080 a b c^2 x-2048 a c^6 x-2560 a^2 c^4 x^2+40320 a^3 c^2 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (37520 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{55440 a c^2 \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{5/2}} \]

output

1/55440*((40320*a^4*c^2*x^4-2560*a^3*c^4*x^3-2048*a^2*c^6*x^2+114240*a^2*b 
*c^2*x^2-1920*a*b*c^4*x-1024*b*c^6+3465*a*b^2*x+32760*b^2*c^2)*(c+(a*x+(a^ 
2*x^2+b)^(1/2))^(1/2))^(1/2)+(2240*a^3*c^3*x^3+1536*a^2*c^5*x^2+2048*a*c^7 
*x+38640*a*b*c^3*x+768*b*c^5-2310*b^2*c)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+ 
(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(a^2*x^2+b)^(1/2)*((40320*a^3*c^2*x^3 
-2560*a^2*c^4*x^2-2048*a*c^6*x+94080*a*b*c^2*x-640*b*c^4+3465*b^2)*(c+(a*x 
+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(2240*a^2*c^3*x^2+1536*a*c^5*x+2048*c^7+3 
7520*b*c^3)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2) 
)^(1/2)))/a/c^2/(a*x+(a^2*x^2+b)^(1/2))^(3/2)-1/16*b^2*arctanh((c+(a*x+(a^ 
2*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(5/2)
 

3.31.45.2 Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.92 \[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (105 b^2 \left (312 c^2-22 c \sqrt {a x+\sqrt {b+a^2 x^2}}+33 \left (a x+\sqrt {b+a^2 x^2}\right )\right )+16 b c^2 \left (-64 c^4+48 c^3 \sqrt {a x+\sqrt {b+a^2 x^2}}-40 c^2 \left (3 a x+\sqrt {b+a^2 x^2}\right )+420 a x \left (17 a x+14 \sqrt {b+a^2 x^2}\right )+35 c \sqrt {a x+\sqrt {b+a^2 x^2}} \left (69 a x+67 \sqrt {b+a^2 x^2}\right )\right )+64 c^2 \left (a x+\sqrt {b+a^2 x^2}\right ) \left (630 a^3 x^3+32 c^5 \sqrt {a x+\sqrt {b+a^2 x^2}}+8 a c^3 x \left (-4 c+3 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+5 a^2 c x^2 \left (-8 c+7 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )\right )}{\left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}-3465 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{55440 a c^{5/2}} \]

input

Integrate[Sqrt[b + a^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a* 
x + Sqrt[b + a^2*x^2]]],x]
 

output

((Sqrt[c]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]*(105*b^2*(312*c^2 - 22*c 
*Sqrt[a*x + Sqrt[b + a^2*x^2]] + 33*(a*x + Sqrt[b + a^2*x^2])) + 16*b*c^2* 
(-64*c^4 + 48*c^3*Sqrt[a*x + Sqrt[b + a^2*x^2]] - 40*c^2*(3*a*x + Sqrt[b + 
 a^2*x^2]) + 420*a*x*(17*a*x + 14*Sqrt[b + a^2*x^2]) + 35*c*Sqrt[a*x + Sqr 
t[b + a^2*x^2]]*(69*a*x + 67*Sqrt[b + a^2*x^2])) + 64*c^2*(a*x + Sqrt[b + 
a^2*x^2])*(630*a^3*x^3 + 32*c^5*Sqrt[a*x + Sqrt[b + a^2*x^2]] + 8*a*c^3*x* 
(-4*c + 3*Sqrt[a*x + Sqrt[b + a^2*x^2]]) + 5*a^2*c*x^2*(-8*c + 7*Sqrt[a*x 
+ Sqrt[b + a^2*x^2]]))))/(a*x + Sqrt[b + a^2*x^2])^(3/2) - 3465*b^2*ArcTan 
h[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(55440*a*c^(5/2))
 

3.31.45.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {a^2 x^2+b} \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sqrt {a^2 x^2+b} \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}dx\)

input

Int[Sqrt[b + a^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sq 
rt[b + a^2*x^2]]],x]
 

output

$Aborted
 

3.31.45.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.31.45.4 Maple [F]

\[\int \sqrt {a^{2} x^{2}+b}\, \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}d x\]

input

int((a^2*x^2+b)^(1/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1 
/2))^(1/2))^(1/2),x)
 

output

int((a^2*x^2+b)^(1/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1 
/2))^(1/2))^(1/2),x)
 

3.31.45.5 Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.18 \[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\left [\frac {3465 \, b^{2} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) + 2 \, {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} + 37520 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 385 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 1155 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} + 8400 \, a^{2} c^{3} x^{2} - 32760 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 693 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 5712 \, a c^{3} x - 693 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{110880 \, a c^{3}}, \frac {3465 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} + 37520 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 385 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 1155 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} + 8400 \, a^{2} c^{3} x^{2} - 32760 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 693 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 5712 \, a c^{3} x - 693 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{55440 \, a c^{3}}\right ] \]

input

integrate((a^2*x^2+b)^(1/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2 
+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
 

output

[1/110880*(3465*b^2*sqrt(c)*log(2*(a*sqrt(c)*x - sqrt(a^2*x^2 + b)*sqrt(c) 
)*sqrt(a*x + sqrt(a^2*x^2 + b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))) - 
2*(a*c*x - sqrt(a^2*x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2 + b)) + b) + 2*(20 
48*c^8 + 1120*a^2*c^4*x^2 + 37520*b*c^4 + 6*(128*a*c^6 + 385*a*b*c^2)*x + 
2*(384*c^6 + 560*a*c^4*x - 1155*b*c^2)*sqrt(a^2*x^2 + b) - (1024*c^7 + 840 
0*a^2*c^3*x^2 - 32760*b*c^3 + 5*(128*a*c^5 + 693*a*b*c)*x + 5*(128*c^5 - 5 
712*a*c^3*x - 693*b*c)*sqrt(a^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2 + b)))*s 
qrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^3), 1/55440*(3465*b^2*sqrt(-c 
)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/c) + (2048*c^8 + 
 1120*a^2*c^4*x^2 + 37520*b*c^4 + 6*(128*a*c^6 + 385*a*b*c^2)*x + 2*(384*c 
^6 + 560*a*c^4*x - 1155*b*c^2)*sqrt(a^2*x^2 + b) - (1024*c^7 + 8400*a^2*c^ 
3*x^2 - 32760*b*c^3 + 5*(128*a*c^5 + 693*a*b*c)*x + 5*(128*c^5 - 5712*a*c^ 
3*x - 693*b*c)*sqrt(a^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + 
sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^3)]
 

3.31.45.6 Sympy [F]

\[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {a^{2} x^{2} + b}\, dx \]

input

integrate((a**2*x**2+b)**(1/2)*(a*x+(a**2*x**2+b)**(1/2))**(1/2)*(c+(a*x+( 
a**2*x**2+b)**(1/2))**(1/2))**(1/2),x)
 

output

Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b)))*sqrt(a*x + sqrt(a**2*x* 
*2 + b))*sqrt(a**2*x**2 + b), x)
 

3.31.45.7 Maxima [F]

\[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int { \sqrt {a^{2} x^{2} + b} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \,d x } \]

input

integrate((a^2*x^2+b)^(1/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2 
+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
 

output

integrate(sqrt(a^2*x^2 + b)*sqrt(a*x + sqrt(a^2*x^2 + b))*sqrt(c + sqrt(a* 
x + sqrt(a^2*x^2 + b))), x)
 

3.31.45.8 Giac [F(-2)]

Exception generated. \[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a^2*x^2+b)^(1/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2 
+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con 
st gen &
 

3.31.45.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,\sqrt {a^2\,x^2+b}\,\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}} \,d x \]

input

int(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(1/2)*(c + ((b + a^2*x 
^2)^(1/2) + a*x)^(1/2))^(1/2),x)
 

output

int(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(1/2)*(c + ((b + a^2*x 
^2)^(1/2) + a*x)^(1/2))^(1/2), x)